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arXiv:1610.08612 · Physical review. D/Physical review. D.

Multiresolution decomposition of quantum field theories using wavelet bases

Tracie Michlin, Wayne Nicholas Polyzou, Fatih Bulut

Published May 1, 2017• 11 citations
arXivHTMLPDF

Abstract

We investigate both the theoretical and computational aspects of using wavelet bases to perform an exact decomposition of a local field theory by spatial resolution. The decomposition admits natural volume and resolution truncations. We demonstrate that flow equation methods can be used to eliminate short-distance degrees of freedom in truncated theories. The method is tested on a free scalar field in one dimension, where the spatial derivatives couple the degrees of freedom on different scales, although the method is applicable to more complex field theories. The flow equation method is shown to decouple both distance and energy scales in this example. The response to changing the volume and resolution cutoffs and the mass is discussed.

DOI
10.1103/physrevd.95.094501

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We investigate both the theoretical and computational aspects of using wavelet bases to perform an exact decomposition of a local field theory by spatial resolution. The decomposition admits natural volume and resolution truncations. We demonstrate that flow equation methods can be used to eliminate short-distance degrees of freedom in truncated theories. The method is tested on a free scalar field in one dimension, where the spatial derivatives couple the degrees of freedom on different scales, although the method is applicable to more complex field theories. The flow equation method is shown to decouple both distance and energy scales in this example. The response to changing the volume and resolution cutoffs and the mass is discussed.
This thesis investigates the use of Daubechies wavelets to separate scales in local quantum field theory. Field theories have an infinite number of degrees of freedom on all distance scales. Quantum field theories are believed to describe the physics of subatomic particles. These theories have no known mathematically convergent approximation methods. Daubechies wavelet bases can be used separate degrees of freedom on different distance scales. Volume and resolution truncations lead to mathematically well-defined truncated theories that can be treated using established methods. This work demonstrates that flow equation methods can be used to block diagonalize truncated field theoretic Hamiltonians by scale. This eliminates the fine scale degrees of freedom. This may lead to approximation methods and provide an understanding of how to formulate well-defined fine resolution limits.
Flow equation methods, more generally known as Similarity Renormalization Group (SRG) techniques, were developed to address multiscale problems where multiple length or energy scales contribute simultaneously. In this Thesis, we formulate the flow equation method within a wavelet-based framework and apply it to study scale (resolution) separation in a two-dimensional scalar field theory. We demonstrate that the flow systematically block-diagonalizes the Hamiltonian with respect to wavelet resolution, achieving improved truncation compared to earlier studies. Using a model of two real scalar fields coupled through a quadratic interaction, we show that the flow equations effectively suppress couplings between low- and high-resolution degrees of freedom. This provides a clear mechanism for isolating low-resolution physics and offers insight into the construction of effective Hamiltonians using a wavelet-based flow equation approac
We study the low energy dynamics of a system of two coupled real scalar fields in 1 + 1 dimensions using the flow equation approach of the similarity renormalization group in a wavelet basis. This paper presents an extension of the work by Michlin [Multiresolution decomposition of quantum field theories using wavelet bases, .] at one resolution higher. We also present the analysis of a model of two scalar fields coupled through a generally quadratic interaction in 1 + 1 dimensions using wavelet-based flow equations. We demonstrate that the specifically chosen generator flows the Hamiltonian into a block diagonal form with each diagonal block being associated with a fixed resolution. The wavelet basis is known to transform the scalar field theory into a model of coupled localized oscillators, each of which is labeled by location and resolution indices. The chosen interaction represents the coupling between two types of oscillators at the same location and resolution index. There is no coupling between oscillators across locations and resolutions. We show that wavelet-based flow equations carry out scale separation while maintaining the interactions between the two scalar fields at each resolution. The effective Hamiltonian associated with the coarsest resolution is shown to correctly reproduce the normal mode frequencies of the model.
We propose an effective Hamiltonian formulation of quantum field theories using a Daubechies wavelet basis in position space. Combined with flow-equation methods of the similarity renormalization group (SRG), this approach provides an efficient framework for analyzing quantum field theories by reducing the dimensionality of the Hamiltonian and systematically decoupling degrees of freedom across scales. As an application, the free scalar field theory has been reformulated within this framework to calculate the low-lying energy spectrum of the theory. These basis elements are known to transform the free scalar field theory into a theory of coupled localized oscillators, each of which is labeled by a location and a resolution index. In this representation, the Hamiltonian is naturally organized into fixed-resolution blocks, alongside blocks associated with the interactions between different resolutions. To decouple the different resolution modes and obtain a block diagonalized Hamiltonian with each block associated with a fixed resolution, the flow equation approach of SRG is applied. Finally, we demonstrate that with increasing resolution, the low-energy spectrum can be extracted from the effective lowest-resolution block of the Hamiltonian, leading to a significant reduction in computational cost.
Fractal structures naturally emerge in quantum systems whose initial states exhibit spatial discontinuities, a phenomenon first identified by Berry in the paradigmatic case of a particle confined in an infinite potential well. While previous analyses of quantum fractals have mainly relied on spectral decompositions and geometric scaling arguments, their quantitative characterization often depends on scale choices and truncation effects. Here we present a wavelet-based multiresolution framework that enables a direct and assumption-free quantification of quantum fractality. Fractal dimensions are extracted from the scale-dependent distribution of wavelet energies, without invoking prior power-law hypotheses. The method is applied to space and time quantum fractals arising in confined dynamics, as well as to dynamical curves generated by the associated quantum probability flux. These flux-driven trajectories provide a natural space--time parametrization of the underlying fractal structure and yield scaling properties fully consistent with Berry's predictions for space--time fractals. The resulting fractal dimensions are shown to be robust with respect to the choice of wavelet family, numerical cutoffs, and system parameters. Beyond validating earlier conjectures, the present framework offers a unified and computationally efficient tool for the multiscale analysis of quantum fractality in confined and interference-driven quantum dynamics. That is, it provides an operational, scale-adaptive criterion that unifies the characterization of space, time, and space--time quantum fractals within a single, hypothesis-free approach.